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Mathematical proof reveals magic of Ramanujan’s genius

By Jacob Aron

PROOFS are the currency of mathematics, but Srinivasa Ramanujan, one of the all-time great mathematicians, often managed to skip them. Now a proof has been found for a connection that he seemed to mysteriously intuit between two types of mathematical function.

The proof deepens the intrigue surrounding the workings of Ramanujan’s enigmatic mind. It may also help physicists learn more about black holes – even though these objects were virtually unknown during the Indian mathematician’s lifetime.

Born in 1887 in Erode, Tamil Nadu, Ramanujan was self-taught and worked in almost complete isolation from the mathematical community of his time. Described as a raw genius, he independently rediscovered many existing results, as well as making his own unique contributions, believing his inspiration came from the Hindu goddess Namagiri. But he is also known for his unusual style, often leaping from insight to insight without formally proving the logical steps in between. “His ideas as to what constituted a mathematical proof were of the most shadowy description,” said G. H.Hardy (pictured, far right), Ramanujan’s mentor and one of his few collaborators.

His ideas as to what constituted a mathematical proof were of the most shadowy description

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Despite these eccentricities, Ramanujan’s work has often proved prescient. This year is the 125th anniversary of his birth, prompting Ken Ono of Emory University in Atlanta, Georgia, who has previously unearthed hidden depths in Ramanujan’s work, to look once more at his notebooks and letters. “I wanted to go back and prove something special,” says Ono. He settled on a discussion in the last known letter penned by Ramanujan, to Hardy, concerning a type of function now known as a modular form.

Functions are equations that can be drawn as graphs on an axis, like a sine wave, and produce an output when computed for any chosen input or value. In the letter, Ramanujan wrote down a handful of what were then totally novel functions. They looked unlike any known modular forms, but he stated that their outputs would be very similar to those of modular forms when computed for the roots of 1, such as the square root -1. Characteristically, Ramanujan offered neither proof nor explanation for this conclusion.

It was only 10 years ago that mathematicians formally defined this other set of functions, now called mock modular forms. But still no one fathomed what Ramanujan meant by saying the two types of function produced similar outputs for roots of 1.

Now Ono and colleagues have exactly computed one of Ramanujan’s mock modular forms for values very close to -1. They discovered that the outputs rapidly balloon to vast, 100-digit negative numbers, while the corresponding modular form balloons in the positive direction.

Ono’s team found that if you add the corresponding outputs together, the total approaches 4, a relatively small number. In other words, the difference in the value of the two functions, ignoring their signs, is tiny when computed for -1, just as Ramanujan said.

The result confirms Ramanujan’s incredible intuition, says Ono. While Ramanujan was able to calculate the value of modular forms, there is no way he could have done the same for mock modular forms, as Ono now has. “I calculated these using a theorem I proved in 2006,” says Ono, who presented his insight at the Ramanujan 125 conference in Gainesville, Florida, this week. “It is inconceivable he had this intuition, but he must have.”

Figuring out the value of a modular form as it balloons is comparable to spending a coin in a particular shop and then predicting which town that coin will end up in after a year.

Guessing the difference between regular and mock modular forms is even more incredible, says Ono, like spending two coins in the same shop and then predicting they will be very close a year later.

Though Ono and colleagues have now constructed a formula to calculate the exact difference between the two types of modular form for roots of 1, Ramanujan could not possibly have known the formula, which arises from a bedrock of modern mathematics built after his death.

“He had some sort of magic tricks that we don’t understand,” says Freeman Dyson of the Institute for Advanced Study in Princeton, New Jersey.

While modular forms are mostly related to abstract problems, Ono’s formula could have applications in calculating the entropy of black holes (see “The black hole connection“).

So will Ono’s work turn out to be the last of Ramanujan’s contributions? “I’m so tempted to say that,” says Ono. “But I won’t be surprised if I’m dead wrong.”

The black hole connection

A new formula, inspired by the mysterious work of Srinivasa Ramanujan, could improve our understanding of black holes.

Devised by Ken Ono of Emory University in Atlanta, Georgia, the formula concerns a type of function called a mock modular form (see main story). These functions are now used to compute the entropy of black holes. This property is linked to the startling prediction by Stephen Hawking that black holes emit radiation.

“If Ono has a really new way of characterising a mock modular form then surely it will have implications for our work,” says Atish Dabholkar, who studies black holes at the French National Centre for Scientific Research in Paris. “Mock modular forms will appear more and more in physics as our understanding improves.”